Here's a really fun question a friend told me abut. He claims to know the correct answer, and told me the answer, but left proving the answer as an exercise to me. Now, It's been ages since he asked me the question, and he himself seems to have forgotten the solution, so here's the question for the community(I tried searching, but useless):
Two friends go to a pizza shop and purchase one. They decide to make 8 slices of the pizza. Then the waiter comes and challenges them to cut the pizza such that the point where all cutting lines meet is NOT the centre of the pizza, but another point. The friends are jolts of each other and think that one of them will get a bigger area f pizza is this method is used. To satisfy both of them, the waiter himself cuts it in such a way that both get 4 slices, and there cumulative area is equal. How did the waiter do this ?
So, basically, you have to draw chords across the circle such that all of them intersect at a certain point(thats not the center) and then choose 4 pieces equal in area to the other 4.
My friend claims that if we divide the circle as in the image below:
IN this way, or basically any way, the following property holds:
ar(1) + ar(3) + ar(5) + ar(7) = ar(2) + ar(4) + ar(6) + ar(8)
Hence, the two friends can distribute the pizza amongst themselves.
But now, he doesn't remember how to prove this mathematically. He does say that the proof is very easy and was childish and obvious once you've read it, but he is unable to trace the book this was part of now...
So, I turn to you, friends, help me to solve this.
Thanks in advance,
Nib
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